A note on optimal cheap talk equilibria in a discrete state space

A discrete version of Crawford and Sobel's (1982) cheap talk model is considered. Unlike in the continuous case, limiting attention to partitional equilibria is with loss of generality. The need to consider equilibria that are non-partitional complicates the analysis. It is shown that if utility functions are concave and the sender is upwardly biased, then the receiver's optimal equilibrium is necessarily partitional. Based on this result, a simple characterization of the optimal equilibrium for the discrete uniform quadratic case is proposed.